In a separate court filing on Friday, prosecutors in New York said Cohen should serve a “substantial” jail term for the crimes.

Michael Cohen: Trump’s pit bull

Cohen is one of a number of Trump aides being investigated in Special Counsel Mueller’s probe and has been co-operating with his team.

President Trump has repeatedly denied there was any collusion with Russian officials, calling the investigation a “witch hunt”.

What was in the Mueller memo?

It is a government sentencing memorandum on the one charge of lying to Congress.

Cohen had admitted making false statements about a Trump property deal, out of loyalty to the president.

He said he had submitted a false written statement about a Trump Organization plan to build a skyscraper in Moscow. He said negotiations had gone on for longer than he had suggested and that his contacts with the president on the matter was more extensive than he had claimed.

The memo says that although this crime was serious, any sentencing should be served concurrently with the New York prosecutors’ recommendations on other crimes.

Cohen will be sentenced on 12 December.

The memo says Cohen has taken “significant steps to mitigate his criminal conduct”.

The key elements of the memo are about the Russian links. They are that Cohen:

“Provided information about his own contacts with Russian interests during the campaign and discussions with others in the course of making those contacts”

“Also provided information about attempts by other Russian nationals to reach the campaign”

Spoke to “a Russian national who claimed to be a ‘trusted person’ in the Russian Federation who could offer the campaign ‘political synergy’ and ‘synergy on a government level'”. This person proposed a meeting between Individual 1 (previously identified by Cohen as Mr Trump) and the Russian president, saying it could have a “phenomenal” impact “not only in political but in a business dimension as well”. The meeting did not take place

Provided “Russia-related matters core to its investigation”

Provided “relevant and useful information concerning his contacts with connected to the White House during the 2017-2018 time period

What was in the New York prosecutors’ memo?

The New York prosecutors’ submission said: “Cohen, an attorney and businessman, committed four distinct federal crimes over a period of several years.

“He was motivated to do so by personal greed, and repeatedly used his power and influence for deceptive ends.”

In August, Cohen, who had been Mr Trump’s personal lawyer for more than a decade, pleaded guilty to eight criminal charges, including tax evasion, bank fraud and campaign finance violations.

He said he had paid hush money to two women who alleged they had affairs with Mr Trump, at the direction of “the candidate”, implicating Mr Trump.

Undisclosed payments to bury embarrassing stories about a political candidate can be treated as a violation of US campaign finance laws.

While the prosecutors accepted Cohen’s help with the Mueller investigation should be taken into account in sentencing, they said it should only bring a “modest” reduction on the guidelines of four to five years.

What does it all mean?

Analysis by BBC North America reporter Anthony Zurcher

In a one-two punch of court filings, the memos say that Cohen should get credit for providing useful information to investigators. But not too much credit.

Some of the information in the special counsel’s document is already known. The negotiations for a Trump Tower in Moscow, which lasted well into the 2016 presidential campaign, were once again outlined.

But the document drives home the point that Donald Trump was kept informed about the possible deal, which could have netted “hundreds of millions of dollars” and that the negotiations took place “at a time of sustained efforts by the Russian government to interfere with the US election”.

Other details Cohen supplied are largely left to the imagination.

The president is also once again implicated in the campaign finance charges against Cohen. The New York prosecutors state that Mr Trump’s lawyer acted to silence two women who claim they had affairs with Mr Trump in order to “influence the 2016 presidential election”. What’s more, Cohen “acted in co-ordination with and at the direction of Individual 1”.

What all this means is that prosecutors in New York and Washington are claiming the president had implicit knowledge and involvement in a campaign finance crime. He also had knowledge of a massive property deal that involved contacts with Russian government officials while he was running for president in an election that was being targeted for influence by the Russian government.